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A110104
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a(n) is the number of coverings of 1..n by cyclic words of length 3n, such that each value from 1 to n appears precisely twice. That is, the union of all the letters in all of the words of a given covering is the multiset {1,1,2,2,...,n,n}. No repeats of words are allowed in a given covering.
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3
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OFFSET
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0,2
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COMMENTS
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P-recursive.
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LINKS
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FORMULA
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Differential equation satisfied by egf: sum a(n)t^3n/(3n!) {F(0) = 1, (-2+4*t^6+16*t^3)*(d/dt)F(t) + 4*t^4*(d^2/dt^2)F(t) + t^2*(4+12*t^3+t^6)*F(t)}.
Recurrence: {a(0) = 1, (40320 + 328752*n + 1816668*n^3 + 1102248*n^5 + 398034*n^6 + 1818369*n^4 + 1063116*n^2 + 78732*n^7 + 6561*n^8)*a(n) +(508608*n + 161280 + 453600*n^3 + 34992*n^5 + 2916*n^6 + 173340*n^4 + 661104*n^2)*a(n+1) + (12320 + 19980*n + 12096*n^2 + 3240*n^3 + 324*n^4)*a(n+2) - 2*a(n+3), a(1) = 4, a(2) = 3760}.
a(n) ~ 2^n * 3^(4*n + 1/2) * n^(4*n) / exp(4*n). - Vaclav Kotesovec, Oct 24 2023
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EXAMPLE
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a(1)=4: {123, 132} {112, 233} {113, 322} {133, 122}
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MATHEMATICA
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RecurrenceTable[{(40320 + 328752*n + 1816668*n^3 + 1102248*n^5 + 398034*n^6 + 1818369*n^4 + 1063116*n^2 + 78732*n^7 + 6561*n^8) * a[n] + (508608*n + 161280 + 453600*n^3 + 34992*n^5 + 2916*n^6 + 173340*n^4 + 661104*n^2) * a[n + 1] + (12320 + 19980*n + 12096*n^2 + 3240*n^3 + 324*n^4) * a[n + 2] - 2*a[n + 3] == 0, a[0] == 1, a[1] == 4, a[2] == 3760}, a, {n, 0, 15}] (* Vaclav Kotesovec, Oct 24 2023 *)
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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STATUS
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approved
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